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Basic definitions Problem definition: Given a set of items (S) and a given item (i), define a data structure that supports operations such as find/insert/delete i in constant time. A solution: A hashing function h maps a large data set into a small index set. Typically the function involves the mod( ) operation. Example hashing function: Let S be a set of 10,000 employee records. Let LName be the LastName attribute of the employee record. Suppose each array item can hold up to k employee records. Suppose the array is of size N. (Then Nk > 10,000) Given an employee e, h(e) = e.LName.toInteger( ) % N. CSCI 3333 Data Structures3

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Design of a hash function Two concerns: a.The hash function should be simple enough. b.The hash function should distribute the data items evenly over the whole array. Why? – For (a): efficiency – For (b): to avoid collision, and to make good use of array space. CSCI 3333 Data Structures4

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A sample hash function in Java CSCI 3333 Data Structures5 Exercise: What are the respective hash codes of the following strings: Doe, Smith, Stevenson? Suppose tableSize is 10.

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Linear probing Collision: Given the hash function h, h(x) returns a position that is already occupied. Linear probing: – When a collision occurs, search sequentially in the array until an empty cell is found (to insert the new data item). – Wrap around if necessary. Example below. CSCI 3333 Data Structures6

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Linear probing: example h(k,n) = k % n CSCI 3333 Data Structures7

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Linear probing: insert Q: What’s the worst case cost when inserting an item using linear hashing? N ? Q: What’s the average cost? Theorem 20.2 (next page) The performance of the hash table depends on how full the table is. – The load factor (λ) of a hash table is the fraction of the table that is full (between 0 and 1). CSCI 3333 Data Structures8

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Theorem 20.2 The average number of cells examined in an insertion using linear probing is roughly (1+1/(1- λ) 2 )/2, where λ is the load factor. Exercises: 1.When the table is half full, what’s the average cost of inserting an item? 2.How if the load factor is 25%? 3.How if the load factor is 75%? Why? Primary clustering CSCI 3333 Data Structures9 lfaverage cost

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Primary clustering Large blocks of occupied cells are formed in the hash table. Impact? – Any key that hashes into a cluster requires excessive attempts to resolve the collision. – Plus, that insertion increases the size of the cluster. CSCI 3333 Data Structures10

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Linear probing: search/find Find(k): If the data item k cannot be found at the h(k) position, search sequentially until either k is found or an empty cell is reached; in the latter case k does not exist in the array. Cost? Theorem 20.3 The average number of cells examined in an unsuccessful search using linear probing is roughly (1+1/(1- λ) 2 )/2. The average number of cells examined in a successful search using linear probing is roughly (1+1/(1- λ))/2. CSCI 3333 Data Structures11

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Quadratic probing Goal: To eliminate the primary clustering problem of linear probing Strategy: by examining certain cells away from the original probe point when a collision occurs using F(i) = i 2 Let H = h(k) = hash (k, n). If H is occupied and not equal to k, search H+1, H+2 2, H+3 2, …, until found or all possible locations are exhausted. CSCI 3333 Data Structures13

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Quadratic probing: example CSCI 3333 Data Structures14

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Theorem 20.4 When quadratic probing is used, a new element can always be inserted when the following prerequisites are met: 1)The size of the hash table, M, is a prime number. 2)At least M/2 of the table entries are empty. Overhead of quadratic hashing: The hash table needs to be at least half-empty.  Rehashing of the hash table is needed. CSCI 3333 Data Structures15

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Rehashing A technique to dynamically expand the size of the hash table when, for example, the table is half full in quadratic hashing. Two steps: 1)Create a larger table. 2)Create a new hash function (for example, the table size has changed). 3)Use the new hash function to add the existing data items from the old table to the new table. CSCI 3333 Data Structures16

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Rehashing: example in Java CSCI 3333 Data Structures17

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Separate chaining hashing A more space-efficient hashing method than quadratic hashing. The hash table is implemented as an array of linked list. – The returned value of the hash function points to the linked list where the item is to be inserted or found. Challenge: The linked lists should be kept short. CSCI 3333 Data Structures18

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Separate chaining hashing: analysis Let M be the size of the hash table. Let N be the total number of data items in the hash table. Then, – The average length of the linked list = N/M. – Also called the load factor (lf). Note: different from the load factor (λ) in earlier discussions. – The average number of probes for an insertion = lf. – The average number of probes for an unsuccessful search = lf. – The average number of probes for a successful search = 1+lf/2. CSCI 3333 Data Structures21